Persistence Spheres: Bi-continuous Representations of Persistence Diagrams
This provides a theoretically optimal representation for persistence diagrams, benefiting researchers in topological data analysis and machine learning applications involving complex data types.
The paper tackles the problem of representing persistence diagrams in a way that preserves their geometric structure, introducing persistence spheres as a bi-continuous mapping that ensures stability and fidelity. The result shows state-of-the-art or competitive performance across various tasks, including regression and classification on functional data, time series, graphs, meshes, and point clouds.
We introduce persistence spheres, a novel functional representation of persistence diagrams. Unlike existing embeddings (such as persistence images, landscapes, or kernel methods), persistence spheres provide a bi-continuous mapping: they are Lipschitz continuous with respect to the 1-Wasserstein distance and admit a continuous inverse on their image. This ensures, in a theoretically optimal way, both stability and geometric fidelity, making persistence spheres the representation that most closely mirrors the Wasserstein geometry of PDs in linear space. We derive explicit formulas for persistence spheres, showing that they can be computed efficiently and parallelized with minimal overhead. Empirically, we evaluate them on diverse regression and classification tasks involving functional data, time series, graphs, meshes, and point clouds. Across these benchmarks, persistence spheres consistently deliver state-of-the-art or competitive performance compared to persistence images, persistence landscapes, and the sliced Wasserstein kernel.